Extremal problems with respect to ideal boundary components of an infinite network (Q755933)

The notion of ideal boundary components of an infinite network is introduced with the aid of a sequence of ends as in the potential theory on Riemann surfaces. Related to the set \(P_{A,\alpha}\) of paths from a finite set A of nodes to an ideal boundary component \(\alpha\) and the cuts \(Q_{A,\alpha}\) between A and \(\alpha\), the extremal length \(\lambda_ p=\lambda_ p(P_{A,\alpha})\) and the extremal width \(\mu_ q=\mu_ q(Q_{A,\alpha})\) are defined as usual. A generalized inverse relation: \((\lambda_ p)^{1/p}(\mu_ q)^{1/q}=1\) \((1/p+1/q=1\) and \(1<p<\infty)\) is shown by using the duality theorem on a max-potential problem and a min-work problem for \(P_{A,\alpha}\), and a discrete analogue of the continuity lemma in the theory of extremal length.